91Ó°ÊÓ

A satellite is \(r=10,000 \mathrm{~km}\) from the center of the Earth, traveling perpendicular to the radius vector at \(4 \mathrm{~km} / \mathrm{sec}\). Find \(d \theta / d t\) and \(h\)

Estimate the initial speed of a 100 -meter golf shot hit at \(\alpha=45^{\circ} .\) Is the true \(v_{0}\) larger or smaller, when air friction is included?

Find parametric equations to go around the unit circle with speed \(e^{t}\) starting from \(x=1, y=0 .\) When is the circle completed?

(a) What is the greatest height a ball can be thrown? Aim straight up with \(v_{0}=28\) meters/sec.

Momentum is \(m v\) and its derivative is \(m a=\) force. Angular momentum is \(m H=m \mathbf{R} \times \mathbf{v}\) and its derivative is _____ = torque. Angular momentum is constant under a central force because the _____ is zero.

Find the closest point to the origin on the line \(x=1+t\) \(y=2-t .\) When and where does it cross the \(45^{\circ}\) line through the origin? Find the equation of a line it never crosses.

Find the curvature of a cycloid \(x=a(t-\sin t), y=\) \(a(1-\cos t)\)

(a) How far apart are the two parallel lines \(x=y\) and \(x=y+1 ?(b)\) How far is the point \(x=t, y=t\) from the point \(x=t, y=t+1 ?(c)\) What is the closest distance if their speeds are different: \(x=t, y=t\) and \(x=2 t, y=2 t+1 ?\)

Which vectors follow the same path as \(\mathbf{R}=t \mathbf{i}+t^{2}\) ? The speed along the path may be different. (a) \(2 t \mathbf{i}+2 t^{2} \mathbf{j}\) (b) \(2 t i+4 t^{2} \mathbf{j}\) (c) \(-t \mathbf{i}+t^{2} \mathbf{j}\) (d) \(t^{3} \mathbf{i}+t^{6} \mathbf{j}\)

Find a parametric form for the straight line \(y=m x+b\).